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Full text of "Mathematical And Physical Papers - Iii"

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resistance which increases with the arc more rapidly than if it varied as the first power of the velocity, and so to reduce the observed rate of decrease to what would have been observed in the case of indefinitely small oscillations.
75. In Coulomb's experiments it appeared that the resistance was composed of two terms, one involving the first power, and the other the square of the velocity. If we suppose the same law to hold good in the present case, and denote the amplitude of oscillation at the end of the time t, measured as an angle, by a, we shall obtain
= ..................... (173),
where A and B are certain constants. We must now endeavour to obtain A from the results of observation. Since the substitution for a of a quantity proportional to a will only change the constant B in (173), and the numerical value of this constant is not required for comparison with theory, we may substitute for a the number of lines read off on the scale as entered in Bessel's tables in the columns headed //,.
I have employed four different methods to obtain A from the observed results. The one I am about to give is the shortest of the four, and is sufficiently accurate for the purpose.
The equation (173) gives after dividing by a
=-A- Ba
Now, as has been already observed, the arcs of vibration decrease nearly in geometric progression. If this law were strictly true, we should have
where 0 denotes the initial and 2 the final arc, and T denotes the whole time of observation. We may, without committing any material error, substitute this value of a in the last term of (174). The magnitude of the error we thus commit is not to be judged of merely by the smallness of B. The approximate expression (175) is rather to be regarded as a well-chosen formula of interpo-